Integrand size = 21, antiderivative size = 116 \[ \int \cot (c+d x) (a+b \tan (c+d x))^{3/2} \, dx=-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {(a-i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d} \]
-2*a^(3/2)*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/d+(a-I*b)^(3/2)*arctanh ((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/d+(a+I*b)^(3/2)*arctanh((a+b*tan(d* x+c))^(1/2)/(a+I*b)^(1/2))/d
Time = 0.19 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.96 \[ \int \cot (c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\frac {-2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )+(a-i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )+(a+i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d} \]
(-2*a^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]] + (a - I*b)^(3/2)*Ar cTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + (a + I*b)^(3/2)*ArcTanh[Sq rt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d
Time = 0.95 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.91, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 4056, 3042, 4022, 3042, 4020, 25, 73, 221, 4117, 73, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \cot (c+d x) (a+b \tan (c+d x))^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a+b \tan (c+d x))^{3/2}}{\tan (c+d x)}dx\) |
\(\Big \downarrow \) 4056 |
\(\displaystyle \int \frac {2 a b-\left (a^2-b^2\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx+a^2 \int \frac {\cot (c+d x) \left (\tan ^2(c+d x)+1\right )}{\sqrt {a+b \tan (c+d x)}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {2 a b-\left (a^2-b^2\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx+a^2 \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx\) |
\(\Big \downarrow \) 4022 |
\(\displaystyle a^2 \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-\frac {1}{2} i (a+i b)^2 \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx+\frac {1}{2} i (a-i b)^2 \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a^2 \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-\frac {1}{2} i (a+i b)^2 \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx+\frac {1}{2} i (a-i b)^2 \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx\) |
\(\Big \downarrow \) 4020 |
\(\displaystyle a^2 \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-\frac {(a-i b)^2 \int -\frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}-\frac {(a+i b)^2 \int -\frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle a^2 \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+\frac {(a-i b)^2 \int \frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}+\frac {(a+i b)^2 \int \frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle a^2 \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-\frac {i (a+i b)^2 \int \frac {1}{-\frac {i \tan ^2(c+d x)}{b}-\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}+\frac {i (a-i b)^2 \int \frac {1}{\frac {i \tan ^2(c+d x)}{b}+\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle a^2 \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+\frac {i (a-i b)^{3/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d}-\frac {i (a+i b)^{3/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d}\) |
\(\Big \downarrow \) 4117 |
\(\displaystyle \frac {a^2 \int \frac {\cot (c+d x)}{\sqrt {a+b \tan (c+d x)}}d\tan (c+d x)}{d}+\frac {i (a-i b)^{3/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d}-\frac {i (a+i b)^{3/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {2 a^2 \int \frac {1}{\frac {a+b \tan (c+d x)}{b}-\frac {a}{b}}d\sqrt {a+b \tan (c+d x)}}{b d}+\frac {i (a-i b)^{3/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d}-\frac {i (a+i b)^{3/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {i (a-i b)^{3/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d}-\frac {i (a+i b)^{3/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d}\) |
(I*(a - I*b)^(3/2)*ArcTan[Tan[c + d*x]/Sqrt[a - I*b]])/d - (I*(a + I*b)^(3 /2)*ArcTan[Tan[c + d*x]/Sqrt[a + I*b]])/d - (2*a^(3/2)*ArcTanh[Sqrt[a + b* Tan[c + d*x]]/Sqrt[a]])/d
3.6.16.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[c*(d/f) Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[ b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + I*d)/2 Int[(a + b*Tan[e + f*x])^m*( 1 - I*Tan[e + f*x]), x], x] + Simp[(c - I*d)/2 Int[(a + b*Tan[e + f*x])^m *(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !IntegerQ[m]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(3/2)/((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[1/(c^2 + d^2) Int[Simp[a^2*c - b^2*c + 2*a*b*d + (2*a*b*c - a^2*d + b^2*d)*Tan[e + f*x], x]/Sqrt[a + b*Tan[e + f*x ]], x], x] + Simp[(b*c - a*d)^2/(c^2 + d^2) Int[(1 + Tan[e + f*x]^2)/(Sqr t[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e , f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A/f Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]
Timed out.
hanged
Leaf count of result is larger than twice the leaf count of optimal. 758 vs. \(2 (90) = 180\).
Time = 0.31 (sec) , antiderivative size = 1531, normalized size of antiderivative = 13.20 \[ \int \cot (c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\text {Too large to display} \]
[1/2*(d*sqrt((a^3 - 3*a*b^2 + d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) )/d^2)*log(-(3*a^4 + 2*a^2*b^2 - b^4)*sqrt(b*tan(d*x + c) + a) + (d^3*sqrt (-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) - (3*a^3 - a*b^2)*d)*sqrt((a^3 - 3*a* b^2 + d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))/d^2)) - d*sqrt((a^3 - 3*a*b^2 + d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))/d^2)*log(-(3*a^4 + 2*a^2*b^2 - b^4)*sqrt(b*tan(d*x + c) + a) - (d^3*sqrt(-(9*a^4*b^2 - 6*a^2 *b^4 + b^6)/d^4) - (3*a^3 - a*b^2)*d)*sqrt((a^3 - 3*a*b^2 + d^2*sqrt(-(9*a ^4*b^2 - 6*a^2*b^4 + b^6)/d^4))/d^2)) - d*sqrt((a^3 - 3*a*b^2 - d^2*sqrt(- (9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))/d^2)*log(-(3*a^4 + 2*a^2*b^2 - b^4)*sq rt(b*tan(d*x + c) + a) + (d^3*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) + ( 3*a^3 - a*b^2)*d)*sqrt((a^3 - 3*a*b^2 - d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))/d^2)) + d*sqrt((a^3 - 3*a*b^2 - d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b ^4 + b^6)/d^4))/d^2)*log(-(3*a^4 + 2*a^2*b^2 - b^4)*sqrt(b*tan(d*x + c) + a) - (d^3*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) + (3*a^3 - a*b^2)*d)*sq rt((a^3 - 3*a*b^2 - d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))/d^2)) + 2*a^(3/2)*log((b*tan(d*x + c) - 2*sqrt(b*tan(d*x + c) + a)*sqrt(a) + 2*a)/ tan(d*x + c)))/d, 1/2*(4*sqrt(-a)*a*arctan(sqrt(b*tan(d*x + c) + a)*sqrt(- a)/a) + d*sqrt((a^3 - 3*a*b^2 + d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^ 4))/d^2)*log(-(3*a^4 + 2*a^2*b^2 - b^4)*sqrt(b*tan(d*x + c) + a) + (d^3*sq rt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) - (3*a^3 - a*b^2)*d)*sqrt((a^3 -...
\[ \int \cot (c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right )^{\frac {3}{2}} \cot {\left (c + d x \right )}\, dx \]
\[ \int \cot (c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cot \left (d x + c\right ) \,d x } \]
Timed out. \[ \int \cot (c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\text {Timed out} \]
Time = 4.98 (sec) , antiderivative size = 2260, normalized size of antiderivative = 19.48 \[ \int \cot (c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\text {Too large to display} \]
- atan((32*a*b^15*(a + b*tan(c + d*x))^(1/2)*(a^3/(4*d^2) - (b^3*1i)/(4*d^ 2) - (3*a*b^2)/(4*d^2) + (a^2*b*3i)/(4*d^2))^(1/2))/((a*b^17*16i)/d + (64* a^2*b^16)/d - (a^3*b^15*128i)/d - (128*a^4*b^14)/d + (a^5*b^13*96i)/d - (1 92*a^6*b^12)/d + (a^7*b^11*384i)/d + (a^9*b^9*144i)/d) - (a^2*b^14*(a + b* tan(c + d*x))^(1/2)*(a^3/(4*d^2) - (b^3*1i)/(4*d^2) - (3*a*b^2)/(4*d^2) + (a^2*b*3i)/(4*d^2))^(1/2)*64i)/((a*b^17*16i)/d + (64*a^2*b^16)/d - (a^3*b^ 15*128i)/d - (128*a^4*b^14)/d + (a^5*b^13*96i)/d - (192*a^6*b^12)/d + (a^7 *b^11*384i)/d + (a^9*b^9*144i)/d) - (96*a^3*b^13*(a + b*tan(c + d*x))^(1/2 )*(a^3/(4*d^2) - (b^3*1i)/(4*d^2) - (3*a*b^2)/(4*d^2) + (a^2*b*3i)/(4*d^2) )^(1/2))/((a*b^17*16i)/d + (64*a^2*b^16)/d - (a^3*b^15*128i)/d - (128*a^4* b^14)/d + (a^5*b^13*96i)/d - (192*a^6*b^12)/d + (a^7*b^11*384i)/d + (a^9*b ^9*144i)/d) + (96*a^5*b^11*(a + b*tan(c + d*x))^(1/2)*(a^3/(4*d^2) - (b^3* 1i)/(4*d^2) - (3*a*b^2)/(4*d^2) + (a^2*b*3i)/(4*d^2))^(1/2))/((a*b^17*16i) /d + (64*a^2*b^16)/d - (a^3*b^15*128i)/d - (128*a^4*b^14)/d + (a^5*b^13*96 i)/d - (192*a^6*b^12)/d + (a^7*b^11*384i)/d + (a^9*b^9*144i)/d) + (a^6*b^1 0*(a + b*tan(c + d*x))^(1/2)*(a^3/(4*d^2) - (b^3*1i)/(4*d^2) - (3*a*b^2)/( 4*d^2) + (a^2*b*3i)/(4*d^2))^(1/2)*576i)/((a*b^17*16i)/d + (64*a^2*b^16)/d - (a^3*b^15*128i)/d - (128*a^4*b^14)/d + (a^5*b^13*96i)/d - (192*a^6*b^12 )/d + (a^7*b^11*384i)/d + (a^9*b^9*144i)/d) - (288*a^7*b^9*(a + b*tan(c + d*x))^(1/2)*(a^3/(4*d^2) - (b^3*1i)/(4*d^2) - (3*a*b^2)/(4*d^2) + (a^2*...